I'm starting Physics, and I don't understand why the acceleration along the x-axis is zero for an object thrown near the surface of the Earth. This may be problem specific, but I wouldn't know since I am just beginning.
Because gravity points purely along the y-axis. If you ignore air drag or thrusters or engines (as you would in a projectile problem), there are no forces along the x-direction, and hence no acceleration along the x-direction.
The acceleration in the x-axis can be non-zero.
In most 2-D physics problems using Cartesian coordinates, the convention is that the y-axis is vertical and the x-axis is horizontal (doesn't need to be that way; it is just the convention used).
So, the y-axis is TYPICALLY the name given to the vertical axis. And gravity on a free falling or stationary object TYPICALLY works only in this vertical plane.
If you were to modify your physics problems, say, to make a ball roll freely down a ramp in the earth's gravitational field, you would find that you get acceleration in both the x and y axes.
When you get to 3-D physics problems, the convention changes: Typically x and y are the assigned names for the two horizontal axes of the Cartesian coordinate system, and the z-axis is the vertical one.